Scheduling Cluster Tools With Ready Time Constraints for Consecutive Small Lots

In the semiconductor manufacturing industry, the lot size currently tends to be extremely small, even being only 5-8 wafers, whereas conventional lots have 25 identical wafers. The smaller lot size is made because customers demand extremely small lots, and the number of chips in a large 300 mm wafer has increased. Cyclic scheduling is not applicable for such small lot production because the number of identical work cycles accounts for a small proportion of scheduling as compared to the lengths of the starting and closing transient periods. We therefore examine a new noncyclic scheduling problem of cluster tools for small lot production that considers ready time constraints on the chambers and the robot. The ready times are the epochs when the resources are freed from processing the preceding lot. To solve the scheduling problem, we develop a Petri net model which is a graphical and mathematical method for discrete event dynamic systems. Based on the Petri net model, we also develop a mixed integer programming (MIP) model and a branch and bound (B&B) algorithm for determining an optimal schedule. The B&B algorithm solves lots with up to 25 wafers and eight wafers within 500 s for a single-armed cluster tool and a dual-armed cluster tool, respectively, when three process steps are considered. Therefore, we propose an approximation method for the dual-armed cluster tool that schedules only the first few wafers with the B&B algorithm and the succeeding wafers with a well-known cyclic sequence. From experiments, we conclude that the difference between the approximation method and an optimal makespan is less than 1%. The methods we propose can be used for general noncyclic scheduling problems that can be modeled by Petri nets.